May 27, 2014 - Examples of the collapsing systems of 3, 15 and 24 vortices are ..... algebraic equations was assumed: = â. = f vz vz. Re [ ] Re [ ] 0. (24). 1. 1 2.
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Collapse of n-point vortices in self-similar motion
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The Japan Society of Fluid Mechanics
Fluid Dyn. Res. 46 (2014) 031414 (16pp)
Fluid Dynamics Research doi:10.1088/0169-5983/46/3/031414
Collapse of n-point vortices in self-similar motion Henryk Kudela Department of Numerical Modeling of Flows, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland E-mail: henryk.kudela@pwr.wroc.pl Received 9 July 2013, revised 10 April 2014 Accepted for publication 17 April 2014 Published 27 May 2014 Communicated by Y Fukumoto Abstract
In this paper the initial positions of n-point vortices that collapse in finite time are numerically obtained. Examples of the collapsing systems of 3, 15 and 24 vortices are given. How to obtain the collapse positions of the vortices in the configuration space that are parameterized by the Hamiltonian values is described. A solution for the self-similar collapse trajectories of the vortices is derived. (Some figures may appear in colour only in the online journal) 1. Introduction The study of the dynamics of point vortices in a plane for an ideal incompressible fluid has been a very active research field for a long time. In 1858 Helmholtz (Helmholtz 1858) wrote the paper ‘On integrals of the hydro-dynamical equations, which express vortex motion’ where the well-known equations for the systems of vortices appeared (see (1) below). Kirchhoff showed in his 1877 lectures that the equation for the vortices’ motion may be expressed as a Hamiltonian system (Aref et al 1992). In 1877 Gröbli in his dissertation studied analytically the motion of three vortices and showed that three vortices can collapse to a point in finite time (for the historical background see Aref et al (1992)). Unfortunately, the works of Gröbli were completely forgotten. A three-vortex system is integrable since it poses the three independent integrals that have commuting Poisson brackets (Aref 2007). Liouvilleʼs theorem of classical mechanics assures that in such a case the Hamiltonian system is integrable and can be explicitly reduced to the quadratures (Kozlov 2003). Yet in Batchelorʼs book (Batchelor 1967) it was written that the detail of the motion of three vortices is not evident. The analytical results of Gröbli were rediscovered by Novikov (1975) and 0169-5983/14/031414+16$33.00 © 2014 The Japan Society of Fluid Mechanics and IOP Publishing Ltd Printed in the UK
1
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H Kudela
independently by Aref (1979). The collapse of three vortices in self-similar motion was shown by Aref (1979) and Novikov and Sedov (1979). Novikov and Sedov (1979) gave special, limited examples for the collapse of four and five vortices. The collapse of the vortices belongs to one of the most interesting problems related to the dynamics of vortices. The problem represents the great interest shown in the theoretical investigation of turbulent motion. The collapsing system of vortices changes the scales of the motion that is the fundamental feature of the turbulence. It is one of the scenarios related to the loss of the uniqueness of Eulerʼs equations in three dimensions. We must remember that the point vortices are represented by a set of rectilinear vortex lines in three dimensions. The collapsing solution in two dimensions (2D) becomes a three-dimensional collapsing solution, despite the fact that vorticity is extended to infinity. The point vortex model provides a useful model to many interesting physical systems (Aref 2007) and is the basis for the vortex particle method, a very efficient numerical method for the solution of the Euler as well as the Navier–Stokes equations (Cottet and Koumoutsakos 2000, Kudela and Malecha 2009). It is worth indicating that in atmospheric physics, collapsing vortices can be seen as the process that leads to the formation of a large atmospheric vortex like a cyclone. In this paper we have demonstrated numerically the collapse of the vortices to a point in finite time for any number of vortices. To the best of the authorʼs knowledge, it is the first time such a demonstration has taken place. 2. Equations of motion of system vortices We start by recalling the equations of motion of the vortices system. Let us assume that there are n-point vortices on the plane with distinct positions z = ( z1, z2, … , z n ) ∈ n , zk = xk + iyk and circulations (intensities) Γ1, Γ2, … , Γn , each Γj ∈ ⧹0 . The differential equation that describes the motion of the system of n vortices is (Newton 2001, Kochin et al 1965) n dzk ( t ) i 1 = vk ( z ( t ) ) = , ∑ ′ Γj dt 2π j = 1 zk − z j
(1)
where the prime on the summation indicates omission of the term with j = k. For the system (1) one can prove the following identities (Kochin et al 1965): n
∑Γkvk = 0,
(2)
k=1 n
∑Γkzkvk = k=1 n
∑Γkvkvk = k=1
i ∑ΓΓ k j, 2π k > j
(3)
ΓΓ d ∑ k j ln ( zk − zj). dt k > j 2π i
(4)
Using these identities, one can prove for system (1) the following integrals (Kochin et al 1965): n
H=−
n
1 ∑ ∑ ΓΓ j k ln rjk = const., 2π j = 1 k , k ≠ j
2
rjk = z j − zk
(5)
Fluid Dyn. Res. 46 (2014) 031414
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n
M = Mx + iMy =
n
∑Γj xj + i∑Γj yj = const. j=1
n
V=
⎛ dyk
∑Γk ⎜ xk k=1
⎝
dt
− yk
(6)
j=1
dxk ⎞ 1 ⎟= ∑ΓΓ k j , = const. dt ⎠ 2π k > j
(7)
n
S=
∑Γj ( xj2 + yj2) = const.
(8)
j=1
The invariant (5) is a Hamiltonian and as shown by Kirchhoff, the system (1) can be expressed in the Hamiltonian formulation (Borisov and Mamaev 2005, Newton 2001):
Γk
xk ∂H = , dt ∂yk
Γk
yk dt
=−
∂H . ∂xk
(9)
The invariant (6) is called a linear impulse, (8) is called an angular impulse and (7) is named a virial (Aref 2007). It is possible to create one more invariant as a linear combination of (6) and (7) that plays a role in the study of the self-similar collapse of vortices (Aref 1979, Kimura 1987, Novikov and Sedov 1979)
L=
2 2 2 ∑ΓΓ k jrkj = σS − ( Mx + My )
(10)
k>j
where
σ=
∑Γk = const.
(11)
k
Invariant L is expressed in terms of the mutual separation of the vortices rjk . It is clear that when a system collapses all rij → 0, so L = 0 is the necessary condition for the vortex collapse (Borisov and Mamaev 2005, p 66). Our interest here is to find numerically the initial positions of vortices that collapse in finite time. As was remarked in Aref (2010), due to the invariants of motion it seems that collapse contradicts the intuitive characteristic that the distance between any two vortices can never be much less than the smallest distance between any pair of vortices initially (Batchelor 1967). But as explored for n = 3 in Aref (1979, 2010), Novikov and Sedov (1979), and Kimura (1987), for self-similar motion and some additional conditions for intensities of vortices, the three vortices can collide in the center of vorticity in finite time. Here it will be demonstrated that such a collapse is possible for any n ⩾ 3. 3. Self-similar motions of n vortices in the plane The dynamic of the n-vortices system is determined by the collection of n intensities Γk , k = 1, … , n and the initial position of the vortices zk (0). It is worth taking note that transformation of variable z to z′ by z′ = az + b in (1) leads only to the division of all intensities Γk by ∣a∣ and does not change the shape of the trajectories (Newton 2001, Synge 1949), but only the time t scales to t a 2 . The shapes of trajectories are invariant to the translations, rotations and dilatations. One can call the given configuration 3
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[ z ] = ( z1, z2, … , z n ) and [ z′] as equivalent [ z ] ∼ [ z′] if they differ by the transformation zi′ = azi + b for some a ∈ C and b ≠ 0 (OʼNeil 1987). Further it will be assumed that σ ≠ 0 and that the center of vorticity is shifted to the zero of the coordinate system
xc = Mx σ = 0,
yc = My σ = 0.
(12)
It means that the invariant M = Mx + iMy (6) is equal to zero, Mx = 0 and My = 0 . In this paper the following definition of the self-similar collapse motion of the n vortices is assumed (see also OʼNeil (1987)): Definition 1. The system of the n vortices is in self-similar collapse if there exists complex
function λ ( t ) ∈ , λ ( t ) = λr ( t ) + iλi ( t ) that Re ( λ ) = λr ( t ) < 0 and λi ( t ) ≠ 0 for each k and
dzk = vk = λ ( t ) zk , dt
k = 1, 2, … , n .
(13)
It follows from (13) that for t = 0 we have
vk (0) = λ (0) zk (0),
k = 1, 2, … , n
(14)
and it is easy to see we have
(
)
vk ( t ) − vj ( t ) = λ ( t ) zk ( t ) − z j ( t ) ,
k = 1, 2, … , n .
(15)
To find the solution of system (13) we introduce the new variables (r ( t ), φ ( t ) ) and assume the following form of that solution (see Newton (2001), Kimura (1987))
zk = zk (0) r ( t ) eiφ ( t ) ,
r (0) = 1,
φ (0) = 0.
(16)
After inserting (16) into (1) one obtains
⎛ dr dφ ⎞ i ⎜ + i ⎟ eiφ ( t ) zk (0) = ⎝ dt ⎠ dt 2π =
n
∑′ j
Γj re
−i φ ( t )
(z
k
(0) − z j (0)
1 λ (0) vk ( z (0) ) = zk (0). r r
) (17)
Separating the real and imaginary parts on both sides of equation (17) and comparing them to each other, one obtains the differential equation for r(t) and φ( t )
dr 1 = λr (0), dt r
r
dφ 1 = λi (0). dt r
(18)
Direct integration of (18) gives:
r=
2λr (0) t + 1 ,
φ=
λi (0) ln ( 2λr (0) + 1). 2λr (0)
4
(19)
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The solution of equation (16) has the form ⎞ ⎛ λ i (0) ⎜i ln ( 2λ r (0) t + 1) ⎟ 2λ r (0) ⎠
zk ( t ) = 2λr (0) t + 1 e ⎝ = ( 2λr (0) t + 1)
1 + i λ i (0) 2 2λ r (0)
zk (0)
.
(20)
Solution (20) represents the logarithmic spiral. It is easy to check that when t → Tc where
Tc = −
1 , 2λr (0)
(21)
is called a collapse time then zk ( t ) → 0 , r ( t ) → 0 and φ( t ) → +∞ when λi(0) > 0 or φ( t ) → −∞ when λi(0) < 0 . When one finds the positions zj (0) ≠ 0 then λ(0) (15) and collapse time (21) can be calculated. If λr ( t ) > 0 then the vortices system expands. To change the direction of the motion we should change the sign of the circulations of all the vortices to the opposite one (Newton 2001). If the real part of λ( t ) equals zero, λr = 0 , the vortices are in relative equilibrium and the systems rotate as a solid body. In this case the collapse time is infinite, Tc = ∞. Putting the solution zk = zk (0) r ( t ) eiφ ( t ) to the Hamiltonian (5) one obtains
H ( t ) = H (0) −
n
1 ln r ( t ) 2π
∑ ΓΓ j k.
(22)
k, k≠j
The Hamiltonian (5) during the self-similar motion will conserve, when the virial V = 0. 4. Algebraic equation for the initial position of collapse vortices From (13) follows that for the collapse of vortices in self-similar motion we have
vjzk = vk z j,
k , i = 1, … , n
k ≠ j.
(23)
OʼNeil (1987) proved that there are only n − 3 independent equations of the form (23). Specifically, he proved that if (23) is true for example for j = 4, 5, … , n then it is true also for j = 2, 3. The configuration space of n vortices needs 2n real numbers. To find the collapse position we can use only n − 3 equations (23). But due to the fact that we assume that the centre of vorticity is in the origin of the coordinate system we have Mx = 0 and My = 0 . OʼNeil also proved, using the identities (2) and (3) that if the virial is equal V = 0 then so is the angular impulse S = 0. From (10) it is evident that invariant L also equals zero. From what was said above follows that the collapse configuration of vortices can be determined by the common zeros of the functions: Mx = 0 , My = 0 , S = 0 and 2 (n − 3) equations for real and imaginary parts of the complex functions vz 1 k − vk z1 = 0 . As suggested by OʼNeil (1987), one can expect that the collapsing vortices lie on some curves. To get the particular positions of the collapse vortices on those curves we must use a trick. To close the algebraic system we need three additional equations. To reduce the number of unknowns we fix one of the vortex coordinates, z n = ( xn , yn ) hoping that the rest of the vortices will arrange themselves to fulfil conditions for a collapse. Additionally we n include to the systemʼs real part of the identity (2) Re ∑i = 1Γi vi = 0 . This identity does not provide any information about the structure of the collapse position of the vortices but still should be true for any vortices system. The following non-linear system of
(
5
)
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algebraic equations was assumed:
f1 = Re [ v1z 2 ] − Re [ v2z1] = 0
(24)
f2 = Im [ v1z 2 ] − Im [ v2z1] = 0
(25)
.....
.....................................
fn − 6 = Im [ v1z n − 3] − Im [ vn − 3z1] = 0
(26)
f2n − 5 = Mx = 0
(27)
f2n − 4 = My = 0
(28)
f2n − 3 = S = 0
(29)
⎞ ⎛ n f2n − 2 = Re ⎜ ∑Γi vi⎟ = 0. ⎝ i=1 ⎠
(30)
Thus a 2n − 2 non-linear system of equations with fixed z n = ( xn , yn ) is solved using the Newton method. To succeed in a numerical solution of an algebraic non-linear system of equations one should provide a good guess for the initial solution (starting point). An unconstrained optimization procedure—the Levenberg–Marquart algorithm—was used to find that first approximation by minimizing the function 2n − 2
F ( x1, y1, … , xn − 1, yn − 1) =
∑ fi 2 .
(31)
i=1
Through trials in arranging the initial position of the vortices for the optimization procedure, one finds the minimum of function F ( x1, y1, … , xn − 1, yn − 1) that should be equal approximately to F ≈ 10−20 or smaller and that points x1, y1, … , xn − 1, yn − 1 are used as a first guess for the Newton procedure. To check the quality of the solution one should verify if every zi is different (zi ≠ zj , i ≠ j ) and not equal to zero. The value of λ (0) = vi (0) zi (0) has to be the same for each i = 1,…, n. The invariants Mx , My and S should be equal to zero with working numerical precision. It is interesting to ask what the curves where the collapse positions of the vortices lie look like. We can obtain some tentative picture of these curves in the following manner. For the particular solution x1, y1, … , xn , yn the corresponding Hamiltonian H (z (0) ) = H0 can be calculated. By replacing equation (30) by the Hamiltonian HΔ = H0 + ΔH = const . where ΔH is an arbitrarily small number, one can find the new position of collapsing vortices for this value of Hamiltonian HΔ. By enlarging and decreasing the Hamiltonian one can obtain the set of the points which are joined by the lines given by the curves where the collapse vortices lay. All calculations were done by Mathematica®, v9.01. Mathematica has the following required procedures: FindMinimum—to minimise the function (31), FindRoot to solve the system of equation (24) by the Newton method and NDSolve to solve the system of differential equation (1). It is very important to carry the calculations with sufficient digital precision. Mainly high numerical precision is needed to find the minimum of function (31). In Mathematica the digital precision is fixed by passing to the procedure parameter that is called ‘WorkingPrecision’. Using WorkingPrecision → w causes calculations to be done using numbers with w-digit precision (Ruskeepää 2009, p 409). Further, the values for H and Tc will be represented only by 6 digits after the decimal point. 6
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Figure 1. Collapse of three vortices: ( Γ1, Γ2, Γ3) = ( −2, −2, 1), H (0) = −0.038 1743, Tc = 2.739 534 . The fixed third vortex had coordinates ( x3, y3) = (1, 0).
5. Numerical examples 5.1. Collapse of three vortices
To clarify our numerical procedure and check the numerical algorithm, we begin from the well-known problem of a collapse of three vortices (Aref 2010). According to our algebraic system (24) the collapse system of three vortices is determined only by S = 0, Mx = 0 and
(
n
)
My = 0 and Re ∑k = 1Γkvk = 0 . As a fixed position for the third vortex ( x3, y3) = (1, 0) was
chosen. The circulations of the vortices were Γ = ( Γ1, Γ2, Γ3) = (− 2, − 2, 1). The collapse position together with the collapse trajectories are presented in figure 1. The fixed vortex position ( x3, y3) = (1, 0) is marked in figure 1 by the dashed-line circle. The presented trajectories were obtained using the formula (20) as well as by solving the equations (1) using NDSolve with PrecisonGoal and AccuracyGoal equal to MachinePrecision (see Ruskeepää (2009)). The difference between these two solutions is smaller than 10−13. WorkingPrecision = 32 was chosen. The critical time was Tc = 2.739 534, H (0) = − 0.038 1743. The minimum of the function F ( x1, y1, x2 , y2 ) (31) was ∼ 10−43. After using the FindRoot the center of vorticity xc and yc and the angular impulse S had the
value ∼ 10−33. Next the Hamiltonian HΔ = H (0) ± ΔH where ΔH = 0.0001 was used instead of equation (30). That gave the new collapse positions for the vortices. The successive change of Hamiltonian with Δ permitted drawing on the plane the curves for the collapse position of the vortices. The numerical results were presented in figure 2 (on the left). When the Hamiltonian H was decreased to Hkr2 ≈ − 0.220 6356, the vortices numbered 1 and 2 were moved to location 1′ and 2′. Vortices 1′, 2′ and 3 lay on a straight line. The enlargement of Hamiltonian H (0) = − 0.038 174 to H ≈ 10−8 caused the displacement of vortices 1 and 2 to the position 1″ and 2″. The vortices 1″ , 2″ and 3 create an equilateral triangle. For an equilateral triangle and V = 0 the Hamiltonian is exactly equal to zero. The positions 1′, 2′, 3 and 1″, 2″, 3 formed the only relative equilibria which is a well-known fact for the three vortices problem (Aref 2010, Newton 2001, p 73). In figure 3 the dependence of collapse time on Hamiltonian values is presented. One can notice that the curve Tc ( H ) is very steep when H is approaching 7
Fluid Dyn. Res. 46 (2014) 031414
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Figure 2. The curves of the collapse position of vortices while the Hamiltonian is
changed from H1 = −0.220 6356 , the points (1′, 2′) to H2 ≈ 10−8 for the point at points (1″ , 2″). Vortex number 3, ( x3, y3) = (1, 0) was fixed.
Figure 3. The dependence of the collapse time Tc on the value of Hamiltonian H.
the relative equilibria position. For relative equilibrium the collapse time is Tc = ∞. The
(
)
minimal collapse time was Tcmin = 1.570 796 with H Tcmin = − 0.145 8401. Trajectories for the minimal collapse time Tcmin are presented in figure 4. 5.2. Collapse of 15 vortices
Figure 5 presents the collapse trajectories for 15 vortices in the time interval [ 0, Tc − 0.0001], Tc = 33.901 983. Taking some trials from the FindMinimum procedure the WorkingPrecision = 200 was chosen. Trajectories were obtained using the self-similar solution (16). This solution was checked by solving the differential equation (1) using NDSolve. PrecisionGoal and AccuracyGoal were chosen equal to 16. The difference between these two solutions was smaller than 10−13. The collapse time was Tc = 33.901 983 and the Hamiltonian had the value H (0) = − 0.464 567 78. We repeated the procedure that was described in subsection 5.1 to obtain the curves of the collapse positions of the vortices. The Hamiltonian 8
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Figure 4. Trajectories for the minimal collapse time and with the same intensities as in figure 1; Tc = 1.570 796 33 and H (0) = −0.145 8401.
5. Collapse of the 15 vortices; Γ( 1 to 7) = −1, Γ( 8 to 15) = 1 2, H (0) = −0.464 5678, Tc = 33.901 983. The larger diameter points indicate the negative vortices Γ = −1, smaller diameter points mark the positive vortices with Γ = 1 2. Figure
was changed from H1 = − 0.480 527 78 to H2 = − 0.388 927 78. An increment for the Hamiltonian ΔH = 0.0001 was used. The curves of the collapse positions are presented in figure 6. The minimum of the function F ( x1, y2 , … , x14 , y14 ) along the curves was equal to 10−400 and the invariants S, Mx , My were equal to 10−100 . The curves started from the thick points with H1 that were approximately in the relative equilibrium. The numerically obtained collapse time there was Tc1 = 1141.8582. The crosses on the curves mark the initial locations
of the vortices that were used in figure 5. The curves ended for H2 = − 0.388 827 and the 9
Fluid Dyn. Res. 46 (2014) 031414
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Figure 6. The curves for the locations of the collapse position of the vortices when the Hamiltonian is changed from H1 = −0.480 5278 to H2 = −0.388 827. The crosses on the curves mark initial collapse positions for the trajectories present in the figure 5.
collapse time Tc2 = 30.102 991. Continuations of the curves were impossible due to a sudden drop in the solutionʼs accuracy. The angular impulse S changes from ∼ 10−100 to ∼0.0001. We see that the curves that represent the zeroes of the algebraic equations (24) for different Hamiltonian values have very irregular shapes. The curves change directions sharply. The irregular shapes of the curves indicate the very complex form of the function F (31). That is the reason for the high value of the WorkingPrecision parameter in the FindMinimum procedure. In figure 7 the graph of collapse time Tc versus the Hamiltonian values H is presented. Once again we see that the curve is very steep when the value of H approaches the relative equilibrium. Having the dependence of collapse time on Hamiltonian values one can choose the trajectories for any collapse time, for example for the minimal collapse time. The minimal collapse time was Tcmin = 20.005 039 and is marked in figure 7 by the thick point. 5.3. Collapse of 24 vortices
The last example relates to the system of 24 vortices. Three different values were used for the intensities Γi : Γ1 to Γ11 were equal to −1 , Γ12 to Γ18 were equal to 1 42(− 77 − 1309 ) and Γ19 to Γ24 were equal to 1 15 (11 − 1309 ). The σ = − (7/30) (11 + 1309 ) ≈ − 11.008 691. To go through some trials WorkingPrecision = 240 was chosen. The obtained minimum of function F ( x1, y1, … , xn − 1, yn − 1) (see (31)) was ∼ 10−450 . The values of Mx , My and S were numerically equal to zero. The collapsing trajectories, based on (16) are presented in figure 8. Due to the strong interactions of the vortices near the singular point the NDSolve procedure gave approximately the same results as (16) only for the time 0 ⩽ t ⩽ Tc − 0.15 (PrecisionGoal and AccuracyGoal were chosen equal to 16). The collapse time was Tc = 3.036 775, the Hamiltonian had the value H (0) = − 5.904 073. The fixed position of the vortex assumed during the solution of equations (24) is marked by the dashed-line circle, x24 = − 0.156 7927 and y24 = − 1.491 7828. 10
Fluid Dyn. Res. 46 (2014) 031414
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Figure 7. Dependence of the collapse time on the Hamiltonian. The thick point marks the minimal collapse time Tcmin = 20.005 039, H Tcmin = −0.420 7278.
(
)
Figure 8. Collapse of the 24 vortices; Γ( 1 to 11) = −1, Γ( 12 to 18) = 1 42( −77 − Γ( 19 to 24) = 1 15 (11 − 1309 ), H (0) = −5.904 073, Tc = 3.036 775.
1309 ),
In figure 9 the curves for the collapse positions of the vortices is presented. The Hamiltonian was changed from H1 = − 6.953 037 86 to H2 = − 5.900 6628 with steps of ΔH =
(H
1
− H2
)
1000 . The thick point marks the relative equilibrium. In those points the
collapse time was Tc ≃ 2 · 10 4 , H1 = − 6.953 0379. Invariants Mx , My and S along the curves were zero. The curves end with H2 = − 5.900 663 and Tc = 3.276 088. It was impossible to continue the curves by increasing the Hamiltonian H2 due to the sudden loss of solution accuracy. The value of S changed from ∼ 10−400 to ∼0.0001. In figure 10 the dependence of the collapse time on the Hamiltonian values is presented. The minimal collapse time was Tcmin = 2.256 24 with H = − 6.258 717. In figure 11 the
11
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Figure 9. The curves for the locations of the collapse position of the vortices when the Hamiltonian is changed from H1 = −6.953 0379 to H2 = −5.900 663. The thick points mark relative equilibria H1 = −6.953 0379. The fixed vortex is marked by the circle, ( x24 , y24 ) = (−0.156 793, −1.491 783); Tc = 3.036 775. The points largest in diameter
mean the vortices with negative circulation Γ( 19 to 24) = −1 15 (11 + 1309 ), the medium-sized points mean vortices with positive circulation Γ( 12 to 18) = 1 42 (77 + 1309 ) and the smallest points mean the vortices with negative values Γ( 1 to 11) = −1.
Figure 10. Dependence of the collapse time on Hamiltonian values for data as
in figure 8.
solutions are presented for that minimal collapse time. One can observe that the smaller number of the spiral turns around the origin of the coordinate system. It is also interesting to see the instantaneous distribution of the streamlines induced by the collapsing vortices. The relations between the stream function ψ ( x, y ), vorticity ω and velocities (u, v) are
12
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Figure 11. Trajectories with minimal collapse time for the intensities as in figure 8; Tcmin = 2.25624 ; H = −6.258 717.
Δψ ( x , y) = − ω ( x , y)
u=
∂ψ , ∂y
v=−
(32)
∂ψ . ∂x
(33)
The vorticity can be approximated by the sum of the delta Dirac measures n
ω ( x , y) =
∑Γδj ( x − xj) δ ( y − yj).
(34)
j=1
With the stream function ψ one can express as a sum of stream functions for each individual vortex n
ψ ( x , y) =
∑ψj ( x, y), j=1
ψj ( x , y) = −
Γj 2π
ln
(
( x − xj)2 + y − yj
2
).
(35)
The instantaneous pictures of streamlines for t = 0, t = 0.3Tc , t = 0.9Tc and t = 0.99Tc are presented in figure 12. The thick red points indicate the vortices with the positive intensities, the blue points mean the negative vortices. The black square in the middle of the frames marks the center of vortices (0, 0). To visualize the dynamics of the changes in the distribution of streamlines, the streamline ψ = 0 was drawn using a thick line. Dark gray smudges around the vortices indicates a negative gage pressure smaller than −3. The gage pressure was calculated using the Bernoulli law
v2 . (36) 2 The smudges around the vortices are irregular and the shape of the isoline ψ = 0 is deformed strongly. This indicates interactions between the vortices. As vortices go to the collapse point the smudges merge but still in the frame T = 2.030 52 we see the white places inside of the p=−
13
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t=0
3 2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-2
-1
0
1
2
-3
3
t = 2.03052
3
2
1
1
0
0
-1
-1
-2
-2 -3
-2
-1
0
-3
-2
1
2
-3
3
-1
0
1
2
3
2
3
t = 2.23358
3
2
-3
t = 0.676841
3
-3
-2
-1
0
1
Figure 12. The instantaneous pictures of streamlines of collapse vortices for t = 0, t = 0.3Tc , t = 0.8Tc and t = 0.99Tc . The thick line marks the streamline of zero value, ψ = 0 , Tcmin = 2.256 24. The dark gray color means negative pressure of less than −3.
area round by the vortices (‘the silence zone’). In these places the velocity is nearly zero. At the frame T = 0.99Tc the streamlines are circles as if they are created by the single vortex. Due to fact that invariant V = 0 and σ ≠ 0 one can write
⎛
n
⎞2
∑ΓΓ k j = ⎜ ∑Γk ⎟ k>j
⎝ k=1 ⎠
n
−
2
∑ ( Γk )
= 0,
(37)
k=1
which means that collapse vortices do not annihilate their intensities in the collapse center and the resulting vortex has an absolute value larger than of any of the vortices that took part in the collapse process ΓR =
n
∑k = 1Γk2 (Novikov and Sedov 1979). 14
Fluid Dyn. Res. 46 (2014) 031414
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Figure 13. Polar cyclone on the southeastern coast of Greenland on 4 September 2003. Image courtesy of NASA (Descloitres 2003).
It is worth indicating the relevance of the collapse vortices problem with the meteorological phenomenon of cyclone evolution. The similarity is striking when we compare the spirals of collapse trajectories with vortices of the low-pressure system swirls of the cyclone in figure 13. The cyclone is created by the low regions of low atmospheric pressure that can be identified as vortices.
6. Concluding remarks It was numerically demonstrated that the system of finite numbers of linear vortices under suitable initial conditions can collapse to a point with finite time. The first condition for the collapse of vortices is V = 0. This guarantees the constant value of the Hamiltonian H during the self-similar motion. The shift of the vorticity center to the origin of the coordinate system (M = 0) causes S = 0 and L = 0. Using the features of the self-similar motion of collapse vortices one can build the system of algebraic equations for finding the initial collapse positions of the vortices. The collapse vortices relate to a widely discussed problem of the loss of uniqueness of Eulerʼs equation for inviscid flow (Wayne 2011), the transition to turbulence and stochastization (Novikov 1980). It is well-known that under very little regularity of the initial vorticity it can prove the existence of the global, regular solutions of the Euler equation in 2D (Gallagher and Gally 2005). The singular initial conditions given by the distribution of point vortices may lead to the collapse phenomenon and the loss of the uniqueness. But it is known that the point vortex approximations of the initial vorticity distribution converge to the 2D Euler equations (Goodman et al 1990). The regular solution and the singular one that lead to the non-uniqueness seem to be not far from each other. The 2D point-vortex model is a very useful, though simple, approximation to many important physical systems (Wayne 2011, Novikov 1980) and provides interesting physical interpretations to a number of mathematical 15
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concepts and theorems (Aref 2007). The model seems to be relevant for modeling the Earthʼs atmospheric motion, particularly examples such as cyclones (Descloitres 2003). References Aref H 1979 Motion of three vortices Phys. Fluids 22 393–400 Aref H 2007 Point vortex dynamics: a classical mathematics playground Phys. Fluids 48 065401 Aref H 2010 Self-similar motion of three point vortices Phys. Fluids 22 057104 Aref H, Rott N and Thomann H 1992 Gröbliʼs solution of the three-vortex problem Annu. Rev. Fluid Mech. 24 1–20 Batchelor G 1967 An Introduction to Fluid Dynamics (Cambridge: Cambridge University Press) p 531 Borisov A V and Mamaev I S 2005 Mathematical Methods in the Dynamics of Vortex Structures (Moscow-Izhevsk: Institute of Computer Science) p 15 (in Russian) Cottet G-H and Koumoutsakos P D 2000 Vortex Methods. Theory and Practice (Cambridge: Cambridge University Press) p 10 Descloitres J 2003 MODIS Rapid Response Team NASA/GSFC (http://visibleearth.nasa.gov/view. php?id=68992) Gallagher I and Gally T 2005 Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity Math. Ann. 332 287–327 Goodman J, Hou T Y and Lowengrub J 1990 Convergence of the point vortex method for the 2-D Euler equations Comm. Pure Appl. Math. XLIII 415–30 Helmholtz 1858 Über Integrale der hydro-dynamischen Gleichungen, welche den Wirbeldewegungen entrsprechen J. Reine Angew. Math. 55 25–55 Kimura Y 1987 Similarity solution of two-dimensional point vortices J. Phys. Soc. Japan 56 2024–30 Kochin N E, Kibel I A and Roze N V 1965 Theoretical Hydromechanics (Moscow: Fizmatgiz) p 197 (in Russian) Kozlov V V 2003 Dynamical Systems X: General Theory of Vortices (Encyklopedia of Mathematical Science vol 67) (Berlin: Springer) p 135 Kudela H and Malecha Z 2009 Eruption of a boundary layer induced by a 2D vortex patch Fluid Dyn. Res. 41 1–18 Newton P K 2001 The N-Vortex Problem: Analytical Techniques (Berlin: Springer) p 14 Novikov E A 1975 Dynamics and statistics of a system of vortices Zh. Eksp. Teor. Fiz. 68 1868–82 Novikov E A 1980 Stochastization and collapse of vortex systems Ann. New York Acad. Sci. 357 47–54 Novikov E A and Sedov Yu B 1979 Vortex collapse Zh. Eksp. Teor. Fiz. 77 588–97 OʼNeil K A 1987 Stationary configurations of point vortices Trans. Am. Math. Soc. 302 383–425 Palmore J I 1982 Relative equilibria of vortices in two dimensions Proc. Natl Acad. Sci. 79 716–8 Ruskeepää H 2009 Mathematica Navigator, Mathematics, Statistics, and Graphics (Amsterdam: Elsevier) p 113 Synge J L 1949 On the motion of three vortices Can. J. Math. 1 257–70 Wayne C E 2011 Vortices and two-dimensional fluid motion Not. Am. Math. Soc. 58 10–9
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